\begin{figure*}%[t]
\begin{center}
\subfloat[Data Embedding]{
- \begin{minipage}{0.49\textwidth}
+ \begin{minipage}{0.4\textwidth}
\begin{center}
- %\includegraphics[width=5cm]{emb.pdf}
- \includegraphics[scale=0.45]{emb}
+ %\includegraphics[scale=0.45]{emb}
+ \includegraphics[scale=0.4]{emb}
\end{center}
\end{minipage}
\label{fig:sch:emb}
}
-
+\hfill
\subfloat[Data Extraction]{
\begin{minipage}{0.49\textwidth}
\begin{center}
- %\includegraphics[width=5cm]{rec.pdf}
- \includegraphics[scale=0.45]{rec}
+ \includegraphics[scale=0.4]{dec}
\end{center}
\end{minipage}
\label{fig:sch:ext}
\subsection{Security considerations}\label{sub:bbs}
Among the methods of message encryption/decryption
(see~\cite{DBLP:journals/ejisec/FontaineG07} for a survey)
-we implement the Blum-Goldwasser cryptosystem~\cite{Blum:1985:EPP:19478.19501}
+we implement the asymmetric
+Blum-Goldwasser cryptosystem~\cite{Blum:1985:EPP:19478.19501}
that is based on the Blum Blum Shub~\cite{DBLP:conf/crypto/ShubBB82}
pseudorandom number generator (PRNG) and the
XOR binary function.
-It has been proven~\cite{DBLP:conf/crypto/ShubBB82} that this PRNG
+The main justification of this choice
+is that it has been proven~\cite{DBLP:conf/crypto/ShubBB82} that this PRNG
has the property of cryptographical security, \textit{i.e.},
for any sequence of $L$ output bits $x_i$, $x_{i+1}$, \ldots, $x_{i+L-1}$,
there is no algorithm, whose time complexity is polynomial in $L$, and
on many kinds of architectures like FPGAs, smart phones, desktop machines, and
GPUs, we have chosen this edge detector for illustrative purpose.
-%\JFC{il faudrait comparer les complexites des algo fuzy and canny}
+
This edge detection is applied on a filtered version of the image given
as input.
-More precisely, only $b$ most
-significant bits are concerned by this step, where
-the parameter $b$ is practically set with $6$ or $7$.
+More precisely, only $b$ most significant bits are concerned by this step,
+where the parameter $b$ is practically set with $6$ or $7$.
+Notice that only the 2 LSBs of pixels in the set of edges
+are returned if $b$ is 6, and the LSB of pixels if $b$ is 7.
If set with the same value $b$, the edge detection returns thus the same
set of pixels for both the cover and the stego image.
-In our flowcharts, this is represented by ``edgeDetection(b bits)''.
-Then only the 2 LSBs of pixels in the set of edges are returned if $b$ is 6,
-and the LSBs of pixels if $b$ is 7.
-
-
-
+Moreover, to provide edge gradient value,
+the Canny algorithm computes derivatives
+in the two directions with respect to a mask of size $T$.
+The higher $T$ is, the coarse the approach is. Practically,
+$T$ is set with $3$, $5$, or $7$.
+In our flowcharts, this step is represented by ``Edge Detection(b, T, X)''.
Let $x$ be the sequence of these bits.
The next section presents how to adapt our scheme
- when the size of $x$ is not sufficient for the message $m$ to embed.
+with respect to the size
+of the message $m$ to embed and the size of the cover $x$.
+
\subsection{Adaptive embedding rate}\label{sub:adaptive}
Two strategies have been developed in our approach,
-depending on the embedding rate that is either \emph{adaptive} or \emph{fixed}.
+depending on the embedding rate that is either \emph{Adaptive} or \emph{Fixed}.
In the former the embedding rate depends on the number of edge pixels.
The higher it is, the larger the message length that can be inserted is.
Practically, a set of edge pixels is computed according to the
-Canny algorithm with a high threshold.
-The message length is thus defined to be less than
+Canny algorithm with parameters $b=7$ and $T=3$.
+The message length is thus defined to be lesser than
half of this set cardinality.
If $x$ is too short for $m$, the message is split into sufficient parts
and a new cover image should be used for the remaining part of the message.
-
In the latter, the embedding rate is defined as a percentage between the
number of modified pixels and the length of the bit message.
This is the classical approach adopted in steganography.
Practically, the Canny algorithm generates
-a set of edge pixels related to a threshold that is decreasing
+a set of edge pixels related to increasing values of $T$ and
until its cardinality
-is sufficient. Even in this situation, our scheme is adapting
+is sufficient. Even in this situation, our scheme adapts
its algorithm to meet all the user's requirements.
Once the map of possibly modified pixels is computed,
two methods may further be applied to extract bits that
-are really modified.
+are really changed.
The first one randomly chooses the subset of pixels to modify by
applying the BBS PRNG again. This method is further denoted as a \emph{sample}.
Once this set is selected, a classical LSB replacement is applied to embed the
-\subsection{Minimizing distortion with syndrome-trellis codes}\label{sub:stc}
+\subsection{Minimizing distortion with Syndrome-Trellis Codes}\label{sub:stc}
\input{stc}
% but the whole approach can be updated to consider
% the fuzzy logic edge detector.
-% Next, following~\cite{Luo:2010:EAI:1824719.1824720}, our scheme automatically
-% modifies Canny parameters to get a sufficiently large set of edge bits: this
-% one is practically enlarged untill its size is at least twice as many larger
-% than the size of embedded message.
-
-
-
-%%RAPH: paragraphe en double :-)
+For a given set of parameters,
+the Canny algorithm returns a numerical value and
+states whether a given pixel is an edge or not.
+In this article, in the Adaptive strategy
+we consider that all the edge pixels that
+have been selected by this algorithm have the same
+distortion cost \textit{i.e.} $\rho_X$ is always 1 for these bits.
+In the Fixed strategy, since pixels that are detected to be edge
+with small values of $T$ (e.g. when $T=3$)
+are more accurate than these with higher values of $T$,
+we give to STC the following distortion map of the corresponding bits
+$$
+\rho_X= \left\{
+\begin{array}{l}
+1 \textrm{ if an edge for $T=3$} \\
+10 \textrm{ if an edge for $T=5$} \\
+100 \textrm{ if an edge for $T=7$}
+\end{array}
+\right.
+$$.
follows the data embedding approach
since there exists a reverse function for all its steps.
-More precisely, the same edge detection is applied on the $b$ first bits to
+More precisely, let $b$ be the most significant bits and
+$T$ be the size of the Canny mask, both be given as a key.
+Thus, the same edge detection is applied on a stego content $Y$ to
produce the sequence $y$ of LSBs.
If the STC approach has been selected in embedding, the STC reverse
algorithm is directly executed to retrieve the encrypted message.
-This inverse function takes the $H$ matrix as a parameter.
+This inverse function takes the $\hat{H}$ matrix as a parameter.
Otherwise, \textit{i.e.}, if the \emph{sample} strategy is retained,
the same random bit selection than in the embedding step
is executed with the same seed, given as a key.
\begin{center}
\begin{minipage}{0.49\linewidth}
\begin{center}
-\includegraphics[scale=0.20]{Lena.eps}
+\includegraphics[scale=0.20]{lena512}
\end{center}
\end{minipage}
\begin{minipage}{0.49\linewidth}
\end{figure}
The edge detection returns 18,641 and 18,455 pixels when $b$ is
-respectively 7 and 6. These edges are represented in Figure~\ref{fig:edge}.
+respectively 7 and 6 and $T=3$.
+These edges are represented in Figure~\ref{fig:edge}.
When $b$ is 7, it remains one bit per pixel to build the cover vector.
This configuration leads to a cover vector of size 18,641 if b is 7
and 36,910 if $b$ is 6.
\begin{minipage}{0.49\linewidth}
\begin{center}
%\includegraphics[width=5cm]{emb.pdf}
- \includegraphics[scale=0.20]{edge7.eps}
+ \includegraphics[scale=0.20]{edge7}
\end{center}
\end{minipage}
%\label{fig:sch:emb}
\begin{minipage}{0.49\linewidth}
\begin{center}
%\includegraphics[width=5cm]{rec.pdf}
- \includegraphics[scale=0.20]{edge6.eps}
+ \includegraphics[scale=0.20]{edge6}
\end{center}
\end{minipage}
%\label{fig:sch:ext}
}%\hfill
\end{center}
- \caption{Edge detection wrt $b$}
+ \caption{Edge detection wrt $b$ with $T=3$}
\label{fig:edge}
\end{figure}
which bits to change among the 36,910 ones.
In the two cases, about the third part of the poem is hidden into the cover.
-Results with \emph{adaptive+STC} strategy are presented in
+Results with {Adaptive} and {STC} strategies are presented in
Fig.~\ref{fig:lenastego}.
\begin{figure}[t]
\begin{minipage}{0.49\linewidth}
\begin{center}
%\includegraphics[width=5cm]{emb.pdf}
- \includegraphics[scale=0.20]{lena7.eps}
+ \includegraphics[scale=0.20]{lena7}
\end{center}
\end{minipage}
%\label{fig:sch:emb}
\begin{minipage}{0.49\linewidth}
\begin{center}
%\includegraphics[width=5cm]{rec.pdf}
- \includegraphics[scale=0.20]{lena6.eps}
+ \includegraphics[scale=0.20]{lena6}
\end{center}
\end{minipage}
%\label{fig:sch:ext}
150 & \textrm{if} & \vert X_{ij} - Y_{ij} \vert = 2 \\
225 & \textrm{if} & \vert X_{ij} - Y_{ij} \vert = 3
\end{array}
-\right..
+\right.
$$
This function allows to emphasize differences between contents.
+Notice that since $b$ is 7 in Fig.~\ref{fig:diff7}, the embedding is binary
+and this image only contains 0 and 75 values.
+Similarly, if $b$ is 6 as in Fig.~\ref{fig:diff6}, the embedding is ternary
+and the image contains all the values in $\{0,75,150,225\}$.
+
+
\begin{figure}[t]
\begin{center}
\begin{minipage}{0.49\linewidth}
\begin{center}
%\includegraphics[width=5cm]{emb.pdf}
- \includegraphics[scale=0.20]{diff7.eps}
+ \includegraphics[scale=0.20]{diff7}
\end{center}
\end{minipage}
- %\label{fig:sch:emb}
+ \label{fig:diff7}
}%\hfill
\subfloat[$b$ is 6.]{
\begin{minipage}{0.49\linewidth}
\begin{center}
%\includegraphics[width=5cm]{rec.pdf}
- \includegraphics[scale=0.20]{diff6.eps}
+ \includegraphics[scale=0.20]{diff6}
\end{center}
\end{minipage}
- %\label{fig:sch:ext}
+ \label{fig:diff6}
}%\hfill
\end{center}
\caption{Differences with Lena's cover wrt $b$}
\end{figure}
+
